Surface wave data constitute a powerful tool to investigate the upper mantle structure. All the modern
tomographic models of the whole mantle including these data use the same theory to model the surface waveform.
This theory assumes that the lateral variation of the seismic velocities are small in amplitude and vary smoothly.
Under these hyptothesis, two main approximation are done: the first-order Born approximation, where the heterogeneities
are considered as secondary sources in an unperturbed reference wavefield, i.e. only single scattering is taken
into account, and the asymptotic approximation, where the wavenumber of the seismic signal
is much higher than the wavenumber of the structural anomalies, so the wave path can be modeled by a 0-dimension
ray laterally.
The goal of this study is to estimate experimentally the actual theoretical noise
brought by asymptotic methods on mantle models based on surface wave tomography. To do so
we perform a synthetic test using a non-asymptotic formalism, the 'Higher Order Perturbation Theory' (*HOPT*)
(Lognonné and Romanowicz, 1990; Lognonné, 1991; Clévédé and Lognonné, 1996).
This methodology incorporates the effects of back and multiple
forward scattering on the wavefield by summing modes computed to third order of perturbations directly in the 3-D Earth,
and models the sensitivity to scatterers away from the great-circle path.
The asymptotic method used is the 'Non Linear Asymptotic Theory' (*NACT*) (Li and Romanowicz, 1995). This method is optimized
for body wave modeling, but the surface wave modeling also offers a better accuracy in time with respect to
the classical 'Path Average Approximation' (*PAVA*) (Woodhouse and Dziewonski, 1984).

We have designed a test model consisting in two heterogeneities with power up to spherical harmonic degree 12
(Figure 18.1).
Synthetic data are computed using the *HOPT* method. The data set consists in
7849 seismograms corresponding to the real surface-wave data coverage used in the SAW12D model (Li and Romanowicz, 1996),
in the frequency window 2.5mHz-12.5mHz (400s-80s) band-pass with a cosine taper with corner frequencies
4mHz-10mHz (125s-100s). Time windows corresponding to Love waves trains G1 and G2 are selected.
We use these synthetic data as input in the inversion procedure used by Li and Romanowicz (1996), using the *NACT* method.
Maps of the output model is shown on figure 18.2. The heterogeneities are well retreived both
laterally and radially, but slighty spread in both directions, and spurious structure with small amplitude appears over all the model.
The spectral content and rms profile of the models offer a better comparison: figure 18.3 show
that the asymptotic filter yields a loss in energy at depths where the heterogeneities are located, and some smearing radially.
In order to estimate the effect of the path coverage in this loss of resolution, we performed a 'circular' test using *NACT* to compute
synthetic data as input. The result is shown on figure 18.4: the model is almost perfectly retrieved.
We conclude that the loss of resolution shown on figure 18.3 is due to the inability
of the asymptotic method to reproduce the effects of the Fresnel zone of the surface waves. We must point out
that in the present case, where the initial model has only a large wavelength content, the 'theoretical noise' is still
small, and that the tomographic model avalaible using surface wave data can be considered reliable, as long as
they are limited to large scale structure.

The results of this study are extensively presented in

Clévédé et al. (1998) (a theoretical comparison of the *NACT* and *HOPT* methods
can also be found in this paper). We have shown that we are now able to test experimentally the
effect of asymptotic approximations in the framework of surface wave tomography.
The next step is to investigate the limits of these approximations by using input models with a larger spectral content,
and to test higher order asymptotic approximations that allow to take into account amplitude effects (focusing/defocusing)
(Romanowicz, 1987).
A further step will be to use the *HOPT* method in an iterative inversion scheme for the surface waves, coupled with the
*NACT* method for the body waves.

Clévédé, E., and P. Lognonné, Fréchet derivatives of coupled seismograms
with respect to an anelastic rotating Earth, *Geophys. J. Int.*, *124*, 456-482, 1996.

Clévédé, E., C. Mégnin, B. Romanowicz and P. Lognonné, Seismic waveform modeling
and surface wave tomography in a three-dimensional Earth: asymptotic and non-asymptotic approaches,
Submitted to *Phys. Earth. Planets Inter.*, 1998.

Li, X.-D., and B. Romanowicz, Comparison of global waveform inversions with and without considering
cross-branch modal coupling, *Geophys. J. Int.*, *121*, 695-709, 1995.

Li, X.-D., and B. Romanowicz, Global mantle shear velocity model developed using non-linear
asymptotic coupling theory, *J. Geophys. Res.*, *101*, 22245-22272, 1996.

Lognonné, P., Normal modes and seismograms of an anelastic rotating Earth,
*J. Geophys. Res.*, *96*, 20309-20319, 1991.

Lognonné, P., and B. Romanowicz, Modelling of coupled normal modes of the Earth:
the spectral method, *Geophys. J. Int.*, *102*, 365-395, 1990.

Romanowicz, B., Multiplet-multiplet coupling due to lateral heterogeneity: asymptotic effects
on the amplitude and frequency of the Earth's normal modes,
*Geophys. J. R. Astr. Soc.*, *90*, 75-100, 1987.

Woodhouse, J. H., and A. M. Dziewonski, Mapping the upper mantle: three-dimensional modeling of
the Earth structure by inversion of seismic waveforms, *J. Geophys. Res.*, *89*, 5953-5986, 1984.

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